-----Original Message-----
From: Harvey Friedman <friedman at math.ohio-state.edu>
To: fom at math.psu.edu <fom at math.psu.edu>
Date: Saturday, January 15, 2000 2:39 PM
Subject: FOM: Einstein, Godel, Turing, Hardy
Harvey Friedman:
>The recent FOM postings discussing or touching on Einstein as "man of the
>century" include Insall Sun, 9 Jan 2000 19:29, Friedman Thu, 13 Jan 2000
>23:09, Black Fri, 14 Jan 2000 16:20, Friedman Fri, 14 Jan 2000 12:19, Pratt
>Fri, 14 Jan 2000 11:20, Pratt Fri, 14 Jan 2000 13:28, and Steiner Sat, 15
>Jan 2000 19:24.
>>In my view, the most important aspect of what Einstein did (at least with
>respect to relativity theory) is as foundations. He took on issues of great
>g.i.i. (general intellectual interest), analyzed them with great
>imagination and clarity, and formulated his theories in a sufficiently
>clear and convincing way to the physics community. He used whatever
>mathematical tools were appropriate. The fact that his theories were so
>experimentally confirmable adds greatly to the excitement of influence of
>his work.
>>His is an example of the speical effectiveness of the foundational approach
>to science. The same is true of other celebrated figures, especially Godel
>and Turing.
>>To consider Einstein's general theory of relativity as an example of pure
>mathematics is to trivialize the essence of his achievment. By the same
>token, to consider Godel's theorems or Turing's machines as examples of
>pure mathematics is also to trivialize the essences of their achievments.
To me, this is an interesting viewpoint, but I'm wondering where you
would locate the "essence" of Gödel's and Turing's achievements. I'll get
more specific below.
>>I am quite familiar with the counterproductive process of taking work in
>f.o.m. and stripping away the essence, and considering it merely as a
>contribution to mathematics or mathematical logic. This happens all the
>time, but is a completely misguided approach that does not respect the
>essence of what is achieved.
I thought perhaps you might locate the "essence" of G's and H's
achievements somewhere in the field of logic, but I see from the above that
that's out too.
>The foundational approach and foundational aspects of what Einstein,
>Turing, and Godel did completely dwarf any mathematics that is involved.
>The mathematics involved is completely trivial (even if it is highly
>nontrivial) compared to the foundational essence of what they did.
What would that essence be?
>The foundational essence of what these people did can be conveyed with a
>minimal amount of mathematics. The mathematics gets more involved when one
>wants to give full and complete formulations of the basic ideas -
>formulations that need to be given for the proper further development of
>the ideas by professional scientists.
Ok, I can now ask a more specific question: Do you consider that the
typical one-paragraph informal explanation of Gödel's (first) Theorem
adequately captures the "essence" of his contribution? Here's the schema:
* Let sentence G say 'I am not provable'. Now, suppose G is true.
Etc. etc. etc. What if G is false? Etc. etc. etc.
I personally do not like these informal expositions, and I'd be very
surprised if you would think much of them. Otherwise, I'm not sure what
you're getting at when you speak of the essence of (in this case) Gödel's
result. Could you please explain more thoroughly what you take the essence
of the first theorem to be (and, if you wish, the essence of Turing's
achievements)?
>Steiner Sat, 15 Jan 2000 19:24:
>>As most readers of this list will remember, the major
>>criterion for a theory to be mathematics (according to Hardy) is
>>aesthetic.
Harvey:
>This is not a useful view of mathematics, in that only the tiniest portion
>of the educated public will appreciate this aspect of mathematics.
I disagree with this very strongly. I am under the impression that if
the general public had an inkling that mathematics is beautiful, they'd
become more interested in it and treat it with more respect. Occasionally,
in response to hearing someone complain how much they hated math in high
school, I have mentioned that maybe their teacher didn't appreciate the
beauty of mathematics and forced them to do repetitive, boring, useless
exercises. The mention of "beauty" in this context always seems to give
them pause, as if they're now considering the possibility that the teacher
was awful, not the subject. Of course, saying there are aesthetic aspects
to math would only be a start.
[I note that no one on this list (with the exception of one person, who sent
me a private note) responded to the article from the Chronicle of Higher Ed.
that I posted. To me, this poses a genuine threat to math, math logic, and
to whatever is of g.i.i. (supposing Harvey is right that it's outside these
two areas).]
Incidentally, I loved Paul Erdös's talk of the proofs in "God's book."
>It [the view that good math must meet some aesthetic criterion] also
>is too ill defined (at present) to be generally useful for the evaluation
>of and development of mathematics. Adherence to this view causes great
>difficulties for mathematicians.
The view that mathematics has an aesthetic component doesn't need to be
"well-defined" in order for it to have great influence. Apparently your
point is that *if* one criterion of "good math" is that it meet some
aesthetic goal, then that goal would have to be useful for the evaluation
and development of mathematics. I don't think the "if, then" is correct.
That is, I don't see that saying math has an aesthetic component imposes the
requirement that this aesthetic component be useful for the evaluation and
development of math. For example (this example could get me into trouble),
isn't it right to say that one (maybe "the") criterion for determining a
good oil painting is aesthetic, even though just saying that it's aesthetic
may not be useful in the evaluation and development of good oil paintings?
Even if you don't like my example, I think the point I'm trying to make is
clear.
But, assuming the "if then" really is correct (i.e., that *if* there's
is an aesthetic criterion for good math, then that criterion should be
useful), it seems to me it *could* be possible to somehow incorporate
aesthetic considerations more directly into the evaluation and development
of math.
The fact that mathematicians might find it uncomfortable to come right
out and admit that good math is beautiful doesn't seem like a strong point
to me.
I remember hearing Zadeh tell all about his Fuzzy Logic to a bunch of
logicians at a symposium several years ago. It was clear to me at the time
(though, of course, I could have been wrong) that they thought his stuff was
just plain "ugly." It didn't seem to have the "elegance" they demanded of
good math. In the face of their many questions, Zadeh retreated to saying
things like: It isn't a question of whether some element is entirely *in* a
set or not, but whether "it's in the ballpark." I remember these logicians
being very amused by this "ballpark" notion.
Maybe "aesthetic" needs to be explicated. In the Zadeh case, I'm sure
his views were thought to be "inelegant." They also didn't have a kind of
generality and relatedness, which I think are demanded of good math (or
maybe: "real math"). In fact, I tend to think it could be very useful to
try to spell out some of the aesthetic considerations for good math.
Perhaps eventually--ok, maybe I'm going too far with this--philosophers
teaching aesthetics would include mathematics with music and art, as
examples of fields where beauty is an important consideration.
"Euclid alone has looked on beauty bare."
Charlie Silver